In Exercises 5 and 6, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.


A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (a) exactly 7. Identify any unusual events. Explain.

Verified step by step guidance

1. Step 1: Determine if the normal distribution can be used to approximate the binomial distribution. For this, check the conditions: (1) The sample size (n) must be large enough such that both np ≥ 5 and n(1-p) ≥ 5, where n is the number of trials and p is the probability of success. Here, n = 18 and p = 0.37. Calculate np and n(1-p) to verify these conditions.
2. Step 2: If the conditions are satisfied, proceed to approximate the binomial distribution using the normal distribution. The mean (μ) and standard deviation (σ) of the binomial distribution are given by μ = np and σ = √(np(1-p)). Compute these values.
3. Step 3: Apply the continuity correction to account for the discrete nature of the binomial distribution when using the continuous normal distribution. For the probability of exactly 7 successes, use the interval [6.5, 7.5] in the normal distribution.
4. Step 4: Standardize the interval [6.5, 7.5] using the z-score formula: z = (x - μ) / σ, where x is the value of interest, μ is the mean, and σ is the standard deviation. Compute the z-scores for 6.5 and 7.5.
5. Step 5: Use the standard normal distribution table (or a calculator) to find the probabilities corresponding to the z-scores obtained in Step 4. Subtract the smaller probability from the larger probability to find the probability of exactly 7 successes. Sketch the graph of the normal distribution with the shaded region representing this probability.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, it applies to the scenario of selecting undergraduates who prefer to take a job in a different state, where each selection can be viewed as a trial with two outcomes: preferring a job out of state or not.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution is applicable when the number of trials is large, and both the number of successes and failures are sufficiently high (typically np ≥ 5 and n(1-p) ≥ 5). This allows us to use the normal distribution to estimate probabilities for binomial scenarios, simplifying calculations and providing a continuous approximation.

Unusual Events

An unusual event in statistics is typically defined as one that has a low probability of occurring, often less than 5%. In the context of this problem, identifying unusual events involves calculating the probability of selecting exactly 7 undergraduates who prefer a job out of state and determining if this probability falls below the threshold for being considered unusual.